I am far from a mathematician. I took calculus for the first time shortly before my 70th birthday. I am in awe of the giants in the field—Gauss, Euler, Newton, Riemann, Hilbert and von Neumann among others—and do not pretend to have any special talent for abstract symbols or proofs. Quite the contrary! But trying to really understand finance over many years forced me to explore some fundamental ideas in mathematics. One of those ideas—embedded in far more of modern finance than most people realize—is Euler's number, commonly written as e.
At first glance, e looks perplexing: an irrational number, approximately 2.71828, that appears across mathematics, physics, and engineering. What is so special about this particular number with an infinite number of digits to the right of the decimal point? While I am totally at a loss to speak regarding physics, engineering or higher abstract math, in finance, its appearance is easy to understand and grasp. It shows up because finance, while perhaps never explicitly acknowledging this, has come to think about time, growth, and risk as continuous processes.
The easiest place to begin is with the calculation of interest.
Compounding, Taken to Its Limit
Financial assets grow proportionally. That means you earn interest on your current balance. When something grows at a rate proportional to its current size, you're describing exponential growth. Consequently, e makes an immediate appearance since the function ex has a unique property—its rate of change (first derivative) is always equal to itself. That property is often called 'magic' by mathematicians (including my beautiful and brilliant PhD mathematician daughter-in-law)—and why it's inevitable that e appears whenever we think about continuous proportional growth in finance.
Most people encounter compounding early in life, often in the context of a savings account or retirement plan. The idea is straightforward: earn interest not just on your original investment, but also on the interest that has already accrued but not been paid. Compound annually, and you get one result. Compound monthly or daily and end up with a slightly higher amount. You see this all the time when interest rates are being quoted—the annual stated interest rate and the effective rate (given more frequent compounding).
What happens if you keep increasing the frequency of compounding?
As the compounding interval shrinks—quarterly, monthly, daily, hourly—the growth curve approaches a limit. That limit is Euler's number. Assume you have $1. For purposes of illustration and to keep the math simple assume (quite unrealistically) you are earning a rate of 100% compounded annually. At the end of the year, you will have $2. Simple enough. But if you shorten and shorten the compounding period until it is continuous what you will have at the end of the year is $2.718. The familiar formula (1+r)t becomes ert when time is treated as continuous rather than chopped into time-based increments. That is why e is often referred to as the limit of continuous compounding.
There is something revealing about this. Euler's number enters finance not because markets are mystical, but because finance stopped thinking in discrete steps. When you imagine value growing smoothly through time, rather than jumping at fixed intervals, e appears automatically.
Discounting: Growth Run in Reverse
Once you see e in compounding, its role in discounting becomes obvious. Discounting future cash flows is simply compounding run backward. Instead of asking how your money grows over time, we ask how much a future dollar is worth today.
Under continuous discounting, the present value of a future cash flow follows an exponential decay pattern. This formulation makes explicit something that spreadsheets often obscure: time erodes value exponentially, not linearly. A small change in the discount rate can dramatically alter present values when time horizons are long. That is not a modeling quirk—it is a structural feature of exponential decay.
Anyone who has spent time building discounted cash flow models has felt this, even if they never articulated it mathematically. Long-duration assets are extremely sensitive to rates because discounting is an exponential process.
Markets as Continuous Processes
The deeper role of e emerges when we ask how to measure returns over time. When finance professionals calculate investment returns over time, often they move beyond simple percentages and use natural logarithms. If an asset goes from $100 to $150, the log return is ln (150/100) = ln (1.5) ≈ 0.405, or about 40.5%. This might seem needlessly complex, but it has a crucial property: log returns add across time periods.
If you gain 10% one period and 10% the next, your simple returns don't add to 20%—but your log returns do add. This is because with continuous compounding, the final value equals the initial value multiplied by e raised to the total return.
Taking natural logs of both sides turns compounded returns into numbers that add across time. This is why research papers and risk models often speak in terms of log returns and Bloomberg terminals make it easy to compute both percentage and log returns. It's not mathematical showing off—it's the natural way to think about continuous growth. It explains why e is the base of the natural logarithm.
This same mathematics—exponential growth and continuous processes—underlies much of modern finance. The Black-Scholes option pricing model, which assumes that prices grow continuously, relies on e throughout. The normal (Gaussian) distribution, the bell curve underlying most risk models, built on the premise of exponential decay of squared distances, has e embedded in its formula. Once you start looking, you see it everywhere. Not because finance borrowed some exotic mathematical constant, but because the continuous, proportional nature of growth and risk makes e unavoidable.
Why Small Changes Matter More Than You Would Expect
Let's return to our discounted cash flow model—the one you stayed up all night building. You change the discount rate slightly—perhaps by one percent or even just 50 basis points—and the valuation moves far more than feels intuitive. Nothing about the underlying business has changed. The cash flows are the same. The strategy is the same. And yet the number on your computer screen shifts dramatically.
This is not a flaw in the model. It is a feature of how time works in finance.
We tend to think of rates as linear inputs. A one-point change sounds modest, even trivial. But discounting does not operate in straight lines. It compounds continuously through time. Each year's discount compounds on top of the last, which means small differences in rates are magnified as the time horizon lengthens. Over long durations, the effect is profound.
The same intuition shows up elsewhere. Long-duration assets are far more sensitive to interest rates than short-duration ones. Leverage amplifies outcomes much faster than expected. Fees that appear insignificant on an annual basis quietly erode returns over decades. None of this is surprising once you accept that finance is governed by exponential processes rather than linear ones.
Euler's number sits underneath all of this. It is not an abstract curiosity, but the mathematical way of expressing continuous compounding and continuous decay. It explains why time exerts such a powerful influence on financial outcomes—and why intuition, trained on linear experience, so often underestimates its effect.
Closing Thoughts
Finance is not arithmetic but rather exponential and logarithmic. Consequently, Euler's number is front and center. It is true that understanding its central role is not necessary to master finance. It doesn't make someone a better CFO, board member, banker or investor—at least not directly.
But appreciating why it appears in finance can help provide intuitive understanding. It reinforces the idea that time, rates, and growth interact in ways that are nonlinear and consequently, more dramatic. It highlights why small differences matter so much, and why forecasting future outcomes can be a treacherous business. Finally, and perhaps most importantly, it is fascinating and provides one more insight into "Business, Finance and the World We Live In."